1. Interacting Network Elements: Chaos and Congestion Propagation Gábor Vattay Department of Physics of Complex Systems Eötvös University, Budapest, Hungary

4. Web servers Web client

5. Traffic

6. Convergence of technology Internet protocol (IP) takes over The Information has to be cut into packets Packets get a universal IP address and handled by heterogeneous network elements

7. From servers to users User Servers ACK-s

8. The flow

9. Internet

10. Basics of traffic modeling

11. Router (telephone exchange) Incoming phone calls Outgoing phone lines NQ

12. Erlang’s formula (1917) - Analyzed the phone calls in a small danish village and came up with a robust model Number of subscribers: N Number of outgoing lines: Q Call arrival rate [calls/sec] Call holding times [sec] What is the distribution of occupied lines ?

13. 012Q Prob. To have n occupied lines at time t Markovian model for line occupancy n= Poisson distribution

14. On short time scales the process is Brownian

15. Typical internet traffic traces W. E. Leland et al. SIGCOMM 93

16. 1/f noise in ‘ping’ traces I. Csabai, Journal of Physics A27, L417 (1994)

17. Modeling Internet traffic It is harder to smooth out Internet traffic Paxson & Floyd 1995

18. Fractal traffic modeling traffic on a heavily used link [packets/sec] aggregated traffic average+fluctuation average number of packets/sec m mean variance of fluctuations relative variance or time variance

19. for Poisson traffic for Internet traffic H=0.8

20. Hurst exponent on the internet …

21. …and the brain …

22. Mathematical tools

23. Long range dependence (LRD)

24. Internet as a large dynamical system

25. TCP Congestion Control End-to-end principle Round trip time RTT Packet loss detection, time-out, out of sequence packet Packet loss probability Acknowledgement Congestion window: number of unacknowledged packets out in the network

26. Slow-start w=1 each time an ACK arrives two new packets are sent w’ = w + 1 In each round trip time the cwnd doubles Slow-start is terminated after the first packet loss, cwnd is halved w’=[w/2]

27. Congestion avoidance One new packet is sent out at each ACK w’ = w+1/w If cwnd is an integer, then two packets are sent out At each packet loss the cwnd is halved w’=[w/2]

28. Chaos

29. Simplest network model

30. Periodicity Veres & Boda INFOCOM 2000

31. Chaos Veres & Boda INFOCOM 2000

32. Liapunov properties Veres & Boda INFOCOM 2000

34. 3 TCPs with different round trip times Vattay, Marodi, Steger 2002

35. Congestion window evolution

36. Poincarè surface of section Symbols: 1,2,3

37. Fractal dimension of the attractor

38. Symbol sequence probability

39. Topological entropy

40. Basin of attraction

41. 2 TCPs surface of section

42. Topological entropy

43. Interaction of flows

44. Interacting traffic flows Traffic flows crossing the same bottleneck can inherit scaling properties from each other

45. Kenesi, Molnár, Veres, Vattay SIGCOMM 2000

46. Mode locking structure of adaptation Buchta & Vattay 2003 TCP Background (UDP) Bandwidth C

47. Congestion propagation

48. Fukuda &Takayasu 1999 Router-to-router congestion propagation

49. A congestion propagation model Vattay, Steger, Vaderna 2003

50. Simulation results: 10 queues, 10 TCP

51. 50 queues, 1 TCP/queue deffect propagation

52. 10 queues, 5 TCP/queue, web traffic

53. 1TCP/que with initial delay t_d t_d [ms]timespan 0-4000 sec 0 0.01 0.1 0.2 0.3 0.4 0.5 0.6

54. Measuring the speed of propagation: center of mass velocity

55. td[ms] Site/sec 0-0.154421.1-0.140778.2-0.141784.3-0.120683.4-0.059653.5-0.103517.6-0.032555

56. What causes congestion propagation? 1 TCP/queue (ns2) Our fluid model using Baccelli-Hong (2002)

57. Only one assumption is needed: packet loss is more likely for a joining TCP flow at the router